Writing Exponential Functions

Writing Exponential Functions

The Basics

An exponential function with base b is defined by $f\left(x\right)=ab^x$ where a ≠0, b > 0 , b ≠1, and x is any real number.

The base, b, is constant and the exponent, x, is a variable.

More Basics

If all the x-values have a constant difference and all of the y-values have a constant ratio, then the table represents an Exponential Function.

For example, $f\left(x\right)\ =2^x$ is shown in the table to the right.

Example 1

Does the table or equation represent an exponential function? Explain.

Example 2

Find the equation of the exponential function for the table.

Example 3

Find the equation of the exponential function for the table.

Example 4

Find the equation of the exponential function for the table.

Multiple Choice

What is the y-intercept of the function:  $f\left(x\right)=2\left(4\right)^x$  ?

• 2

• 4

•  $\left(4\right)^x$  

• 8

Multiple Choice

What is the y-intercept of the function:  $f\left(x\right)=3\left(\frac{1}{2}\right)^x$  ?

• 3

•  $\frac{1}{2}$  

•  $\left(\frac{1}{2}\right)^x$  

•  $\frac{3}{2}$  

Multiple Choice

What is the Growth/Decay factor of the function:  $f\left(x\right)=2\left(4\right)^x$ ?

• 2

• 8

• 4

• x

Multiple Choice

What is the Growth/Decay factor of the function:  $f\left(x\right)=3\left(\frac{1}{2}\right)^x$  ?

• 3

•  $\frac{1}{2}$  

•  $\left(\frac{1}{2}\right)^x$  

•  $\frac{3}{2}$  

Multiple Choice

Write the equation of the exponential function for the given table.

• $y=3^x$

• $y=12\left(3\right)^x$

• $y=3\left(4\right)^x$

• $y=4\left(3\right)^x$

Multiple Choice

Write the equation of the exponential function for the given table.

• $y=-2\left(\frac{1}{2}\right)^x$
  

• $y=2\left(\frac{1}{2}\right)^x$

• $y=2\left(2\right)^x$

• $y=\frac{1}{2}\left(2\right)^x$

Multiple Choice

Write the equation of the exponential function for the given table.

• $y=2\left(\frac{1}{3}\right)^x$

• $y=-2\left(\frac{1}{3}\right)^x$
  

• $y=\frac{1}{3}\left(2\right)^x$

• $y=2\left(-\frac{1}{3}\right)^x$

Example 5

Write the equation for the graph.

Example 6

Write the equation for the graph.

Example 7

Write the equation for the graph.

Multiple Choice

Write the equation of the exponential function for the given graph.

• y = 2(2)^x

• y = 2(1)^x

• $y=1\left(\frac{1}{2}\right)^x$

• y = 1(2)^x

Multiple Choice

Write the equation of the exponential function for the given graph.

• y = 1(2)^x

• y = 2(1)^x

•  $y=2\left(\frac{1}{2}\right)^x$ 

•  $y=\frac{1}{2}\left(2\right)^x$